Total cross sections of eγ → eXX¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ eX\overline{X} $$\end{document} processes with X = μ, γ, e via multiloop methods

Using modern multiloop calculation methods, we derive the analytical expressions for the total cross sections of the processes e−γ →e−XX¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {e}^{-}X\overline{X} $$\end{document} with X = μ, γ or e at arbitrary energies. For the first two processes our results are expressed via classical polylogarithms. The cross section of e−γ → e−e−e+ is represented as a one-fold integral of complete elliptic integral K and logarithms. Using our results, we calculate the threshold and high-energy asymptotics and compare them with available results.


Introduction
Since the invention of quantum electrodynamics (QED), one of its first touchstones was the calculation of the cross sections of elementary processes, like e + e − → µ + µ − or eγ → eγ. In particular, their total Born cross sections for arbitrary energies invariably appear in any QED textbook. In its vast majority, these results concern the processes with two particles in the initial state and two in the final (2 → 2 processes). In contrast, the total Born cross sections of the 2 → N processes with N > 2 have been paid much less attention. 1 This circumstance is not incidental. It appears that, when there are massive particles in the final state and/or massive propagators in the amplitude, the N -particle phase-space integrals are not so simple to be taken by brute force. From the viewpoint of contemporary multiloop methods, this is no wonder, as the N -particle phase space integral corresponds to the N − 1-loop momentum space integral with bipartite cut. Thus, N 3 case corresponds to (L 2)-loop integrals, which are known to bring much more complexity than one-loop integrals. However, the last 40 years of development of multiloop calculations methods have passed for good reason, and we are now in perfect position to patch this omission. We should not expect too simple results though. It is well-known that  Figure 1. Diagrams contributing to the total cross section of the e − γ → e − µ + µ − process. First three diagrams correspond to C-odd muon pair, while the last two diagrams -to C-even pair. already two-loop integrals can be impossible to express via harmonic polylogarithms [3] and even via generalized polylogarithms [4]. In the present paper we calculate the total cross sections of the processes e − γ → e − XX with X = e − , µ − , γ. These processes have many astrophysical applications, see, e.g., refs. [5][6][7][8][9][10][11], and have been considered in a number of papers [6,9,[12][13][14][15][16][17][18]. Our approach is based on using the optical theorem and Cutcosky rules to express the total cross section via cut diagrams in forward kinematics. 2 The master formula is where s = (p 1 + k 1 ) 2 is the square of total c.m. energy, m e is the electron mass, and Im A e − γ→e − XX is the sum of the propagator-type diagrams with cut e − , X,X lines. Thus, in order to calculate the cross section, we apply to this sum the contemporary multiloop calculation methods: the IBP reduction and the calculation of the master integrals via the differential equations method. Our calculation features a few methods not so widely known in the multiloop community. First, given two differential systems, ∂ x j = M x j and ∂ y j = M y j, we consider the solution of the second system in the form of generalized power series j = α∈S+Z c α (x)y α and show how to systematically obtain the differential equations with respect to x for the finite subset {c α (x)|α ∈ S}. Next, we demonstrate that the contribution of soft photons, as defined by the soft-photon approximation, can be calculated by means of the multiloop techniques, although it does not directly correspond to a sum of conventional Feynman diagrams. Finally, we successfully apply the recently introduced approach to the non-polylogarithmic integrals, based on the construction of -regular basis, ref. [20].
2 Total cross section of the process e − γ → e − µ + µ − Let us first consider the process e − γ → e − µ + µ − . As the muon mass m µ is much larger than m e , we can treat m e as a small parameter. Nevertheless, we can not simply put m e = 0 because the cross section becomes infrared divergent in this limit. Therefore, we keep the electron mass in "large" logarithms and omit the power corrections in m e , so that our final result has the following form: (2.1) Below in this section we will use the units m µ = 1. In order to obtain the above form, we proceed in the following way. We define two LiteRed [21] bases m1 and m2 containing JHEP01(2021)144 the denominators of the corresponding boxed diagrams in figure 1. We add the irreducible numerators D 6 = k 1 · p 3 and D 7 = p 1 · p 3 to basis m1 and D 7 = p 1 · p 3 to m2. We reveal 7 master integrals j = (j 1 , j 2 , . . . , j 7 ) = j m1 0111000 , j m1 −1111000 , j m1 −2111000 , j m1 0111100 , j m2 0111010 , j m2 0111020 , j m2 1111000 and construct the differential systems with respect to m 2 e and s: Since the point m 2 e = 0 is a singular point of the first system, we search for its solution in the form of generalized power series using the Frobenius method, along the lines of ref. [22]. The result has the form j =Ũ c , whereŨ is a fundamental matrix with entries being the generalized power series in m 2 e , and c is a column of constants. Note that we can always redefineŨ by multiplying from the right by any non-degenerate matrix independent of m 2 e . In particular, we can use this freedom to secure that the column of constants c consists of specific coefficients in the asymptotic expansion of master integrals near the point m 2 e = 0. Namely, we secure that where [j k ] m µ e denotes to coefficient in front of m µ e in small-m e asymptotic of j k . The matrix U is found in a routine way, along the lines of ref. [22].
Note that constants c depend nontrivially on s. In order to find the differential system for c with respect to s, we treatŨ in eq. (2.3) as a transformation matrix for the second system in eq. (2.2). Then we obtain Three remarks are in place here. First, since c is independent of m 2 e , so is the matrix M s . Therefore, in order to establish the exact form ofM s , it is sufficient to know only first few terms of generalized power series inŨ . Second, instead of inverting the matrixŨ of truncated generalized power series, we calculateŨ −1 independently from the equation . Finally, equations for constants, corresponding to different fractional powers of m 2 e , decouple, as they should, so thatM s has a block diagonal form In order to find the -forms of the resulting systems, we pass to the variable v = 1 − 4/s. This variable has a simple physical meaning as the maximum muon velocity in JHEP01(2021)144 c.m.f. at a given energy. We have We reduce both systems to -form [23,24] using Libra, ref. [25]. The boundary conditions are fixed by evaluating the small-v asymptotics of j 1 and j 3 : (2.9) Using these boundary conditions, we obtain all c in terms of harmonic polylogarithms. Then we substitute c in eq. (2.3) and obtain a sufficient number of terms in the generalized power series representation for master integrals j. So, our results for master integrals have the form of generalized power series in m 2 e (truncated at some order), whose coefficients are series in expressed via polylogarithms depending on v.

Results
Expressing the cross section via master integrals and substituting our results for the latter, we obtain the total cross section.
Diagrams from figure 1 can be divided into two groups according to C-parity of the muon pair. We present for the reference the contribution of C-odd diagrams (the first three JHEP01(2021)144 diagrams in figure 1) separately:

Cross-section near the threshold
When the muon kinetic energy ∼ v 2 is comparable with m e the derived formulae are inapplicable. It becomes obvious from the small-v asymptotics of eq. (2.10) which reads ln 8v 2 me − 8 3 v 3 , and, therefore, turns negative at v 2 m e . Although this narrow region might be not very relevant for the experiment, let us derive the appropriate expression for the cross section for the sake of completeness. For this purpose we introduce the variable τ via √ s = 2 + (1 + τ )m e . We obtain the differential systems with respect to τ and m e . Then we search for master integrals as a generalized power series in m 2 e , but this time, at fixed τ . We need to fix two nonzero coefficients c = [ . After expressing the cross-section via c we have established that 0 terms of c are sufficient for our purpose. Then we can put = 0, and obtain the system The solution of this system can be expressed via complete elliptic integrals K(−τ /2) and E(−τ /2). We finally get We remind here that τ = √ s−2−me me and that the formula above is valid when 0 < √ s − 2 − m e 1, while eq. (2.10) is valid when m e √ s − 2 − m e . The two formulae agree with each other in the overlapping region m e √ s − 2 − m e 1. In figure 2 we present our exact result (2.10) and the found threshold and high-energy asymptotics. As a cross-check, we present on the same figure also a few points obtained by numerical integration of the differential cross section using the Cuba library [26].

High-energy asymptotics
For the high-energy asymptotics of the total cross-section of e − γ → e − µ + µ − we have (2.14) JHEP01(2021)144 The leading term of this asymptotics has been derived in ref. [15] using the equivalent photon approximation. It coincides with the leading term of our result.

Total cross-section of
For the sake of completeness, let us present also the total cross section of e − γ → e − π + π − , where we consider π-meson as a pointlike scalar particle. We have Note that the leading high-energy asymptotics of this formula, α 3 4 9 ln(s/m e ) − 26 27 , perfectly agrees with the result of ref. [15].

Total cross section of the process e − γ → e − γγ
In this section we present the calculation of the cross section of the process e − γ → e − γγ (double Compton scattering) at arbitrary energies. To avoid the infrared divergences, we JHEP01(2021)144 restrict the integration region by the conditions ω 2 > ω 0 and ω 3 > ω 0 , where ω 2 and ω 3 are the energies of the outgoing photons, and ω 0 is a small cut-off parameter. This restriction is frame-dependent, and we consider two physically relevant frames: the center-of-mass frame and the initial electron rest frame. From the technical point of view, we proceed as follows. Using dimensional regularization, we first calculate the total cross section, which contains the term ∝ 1 . Then we calculate separately the contribution of soft-photon region and subtract it from the total cross section to obtain the physically observed cross section σ e − γ→e − γγ (ω 0 ).

Calculation of the total cross section in dimensional regularization
Using the optical theorem (1.1), the total cross section e − γ → e − γγ can be expressed in terms of cut diagrams shown in figure 3. We set up two LiteRed bases g1 and g2 containing propagators of boxed diagrams in figure 3. We find 14 master integrals, ) we obtain the differential system ∂ s j = M (s, ) j, (3.1) where M is the matrix, rationally depending on s and . Introducing the new variable y = s−1 s+3 , we reduce the system (3.1) to -form using Libra [25]: where S i are some constant matrices. The canonical basis J is connected with j by rational transformation j = T (y, ) J . The general solution of the system (3.2), J = n n J (n) , can be easily written in terms of generalized polylogarithms. We fix the boundary conditions by calculating the coefficients in the asymptotic expansion of the master integrals at the threshold. The only nontrivial coefficient of the leading threshold asymptotic is Using this boundary condition, we find an expression for the master integrals and get the total Born cross sections e − γ → e − γγ in the form where σ 1 , σ 2 can be expressed via generalized polylogarithms with letters 0, ±1, ± i √ 3 . In this expression we omit the terms suppressed by . The 1 term is quite anticipated, it is due to the contribution of soft-photon region. In order to get rid of this term, we have to subtract the contribution related to soft photons.

Soft-photon contribution to e − γ → e − γγ
In a soft-photon approximation the differential cross section factorizes as where dW γ is a soft-photon emission probability (3.6) Here k 3 is the four-momentum of the soft photon. The contribution to the total cross section can be written as Here ω 0 is the maximal energy of soft photon. As we have already mentioned, the infrared cutoff introduces the frame dependence. We calculate the cross section in the center-ofmass frame (cmf) and in the rest frame of the initial electron (rf). Below we present some details of the calculation for the electron rest frame. The soft-photon contribution (3.7) can formally be written as a two-loop integral with cut propagators. Inserting a identity 1 = dωδ (ω − ω 3 ) into (3.7) and rescaling k 3 → ωk 3 we get We calculate the integral on the right side using the differential equations method. This integral should be considered up to the O( 1 ) term.
To determine the basis for IBP reduction, we include propagators from single Compton scattering, scalar products in denominators in (3.7), and cut propagators from δ-functions. We obtain scalar integral

JHEP01(2021)144
where p 2 = p 1 + k 1 − k 2 . We imply that the first four propagators in this integral are cut, and that the last one is the irreducible numerator. Using LiteRed package we reveals the following master integrals j soft 1111000 , j soft 1111010 , j soft 1111100 , j soft 1111200 , j soft 111100−1 , j soft 1111110 . (3.10) Using Libra we find the canonical basis J related to j soft by (3.11) .
To fix the boundary conditions it suffices to calculate the leading asymptotics of j soft 1111000 , Finally, we obtain the contribution of soft-photons in the form (3.14) Note that 1 term in this formula appears to be the same as in eq. (3.4), as it should be. Therefore, in the difference the terms, containing 1 , vanish.

Results
The total Born cross section of the process e − γ → e − γγ, integrated over the kinematic region, in which ω 2,3 > ω 0 , in the electron rest frame

JHEP01(2021)144
For center-of-mass frame we have Here A 2 , A 3 , B 3 , C 3 , D 3 , E 3 are functions which can be expressed via classical polylogarithms. They are defined as follows Let us discuss the limits of applicability of the soft photon approximation used in the present work. For the low-energy region a natural physical requirement is that the cut-off parameter ω 0 should be much smaller than the initial photon energy ω 1 ≈ (s − 1)/2. In terms of the low-energy asymptotics presented below in eq. (3.18) this restriction means that the logarithm in that formula should dominate. For the high-energy region, the requirements for ω 0 is essentially different for the two reference frames we consider. In the center-of-mass frame the infrared cut-off parameter should be small compared to √ s. Thus, it is natural to expect that in the electron rest frame the soft photon approximation breaks down already at ω 0 m e = 1. This is because the photon with energy ∼ m e in the electron rest frame can have energy ∼ √ s when boosted to the center-of-mass frame. This is demonstrated in figure 4, where the results of the present paper are compared with those of numerical integration using the Cuba library [26]. Figure 5 shows the dependence of the cross section on s at ω 0 = 0.01. On the same figure we have shown a few points obtained by numerical integration of the differential cross section using Cuba library. We have also performed the comparison of the obtained results with known numerical and/or approximate results. First, the numerical result of ref. [27] disagrees with our result by a factor of 2. The discrepancy is possibly due to the overlooked Bose symmetry factor 1 2! in ref. [27]. Once this factor is recovered, we find perfect agreement. Numerical results of ref. [28] agree with our results up to s ≈ 30m 2 e . The discrepancy for higher energies is quite expected due to inapplicability of the soft-photon approximation for the parameter ω 0 = ω 1 50 = s−m 2 e 100me chosen in ref. [28].

Asymptotics
Using the expressions for the exact cross sections (3.15) and (3.16) via polylogarithms, it is easy to calculate both the threshold asymptotics and the high-energy asymptotics.
Threshold asymptotics coincides with the non-relativistic asymptotic in ref. [17]. We remind that this formula implies that ω 0 s − 1 1. Note that this asymptotics holds both for electron rest frame and for the center-of-mass frame.
The high-energy asymptotics has the form in the electron rest frame, and in center-of-mass frame. We note that the high-energy asymptotics of the double Compton scattering is obtained for the first time here to the best of our knowledge. The total cross section of the process e − γ → e − e − e + is not expressible via polylogarithms. This is easy to understand because among the relevant master integrals there is an equal mass two-loop sunrise integrals which is a classical example of non-polylogarithmic integrals. Therefore, we will rely here on the approach of ref. [20]. We construct an -regular basis sufficient for the calculation of the cross section. 3 The cut diagrams which contribute to the cross section are shown in figure 6. We define 4 LiteRed bases e1, e2, e3, e4, corresponding to the denominators of the framed diagrams in the same figure. We find 18 master integrals which we choose as follows: Note that the integrals j e1 0101111 and j e3 1101111 can, in principle, be expressed via the integrals of the lower sectors, and can be replaced by, e.g., j e1 0101110 and j e3 1101012 , respectively. However we find our present choice to be convenient as we empirically observe that the IBP reduction of any other integral to our set of masters does not generate inverse powers of in the coefficients. In particular, the total cross section can be expressed as a linear combination of these master integrals with regular coefficients. The differential system for these master integrals is also regular at → 0. Therefore, we can safely put = 0 starting from this point.

JHEP01(2021)144
In order to further simplify the differential system, we use Libra [25] and pass to new master integrals J 1 , . . . , J 18 related to our previous master integrals via

JHEP01(2021)144
Note that, apart from the first two equations, the differential system is strictly triangular and the integrals J 3−12 can be elementary expressed as the iterated integrals depending on J 1 by integrating the corresponding equation. Alternatively, similar to ref. [20], we can express them as one-fold integrals of J 1 and (poly)logarithms. 4 We have only, while eq. (4.11), taking into account the next-to-leading term, would lead to the curve hardly discernible from the exact result: e.g., at √ s = 10m e the difference is a tiny 0.4%. Points show the numerical results of ref. [32].   process. More precisely, we consider i.e., we consider the cross section weighted by the square of invariant mass of the muon pair, the photon pair, and the electron pair for e − γ → e − µ + µ − , e − γ → e − γγ, and e − γ → e − e − e + , respectively. Note that µ 2 γγ dσ e − γ→e − γγ is infrared safe. The numerical results are presented in figure 8. We provide the explicit expressions for the weighted JHEP01(2021)144 cross sections in eq. (5.1) in the supplementary material, see below. Let us note that the leading asymptotics of µ 2 2e − is equal to that of σ e − γ→e − e − e + multiplied by s/2. This can be easily understood on physical ground as follows. At high energies the electron-positron pair is mostly produced with small invariant mass by equivalent photon mechanism. Then we have p 3 ≈ p 4 , p 2 ≈ ( √ s/2, n 2 √ s/2), and p 3 + p 4 ≈ 2p 3 ≈ q( √ s/2, −n 2 √ s/2). Then µ 2 2e − = (p 2 + p 3 ) 2 ≈ 2p 2 · p 3 ≈ s/2. For reader convenience, we provide in the supplementary material the file CSall.m which contains evaluation-ready expressions for the exact cross sections defined in eqs. (2.10), (2.15), (3.15), (3.16), (4.4), and (5.1). When read in from Mathematica session with Get["CSall.m"] the file prints all necessary information about the functions defined therein.
The last but not the least, the calculation presented in this paper has demonstrated a few methods not so widely known in the multiloop community. First, in the calculation of the e − γ → e − µ + µ − cross section we have shown how to obtain the differential equations with respect to one variable (s in our case) for the coefficients of generalized power series with respect to another variable (m 2 e in our case). Next, we have explicitly demonstrated that the contribution of soft photons to the integrated cross section can be calculated by means of the multiloop techniques. Finally, we have successfully applied the recently introduced [20] approach to the non-polylogarithmic integrals, based on the construction of -regular basis. These approaches can be applied in other physically relevant calculations.